Question

Use an appropriate Half-Angle Formula to find the exact value of the expression. Cos(9pi/8)

Answer 1

$\cos^2\dfrac{9\pi}8=\dfrac{1+\cos\frac{9\pi}4}2=\dfrac{1+\cos\frac\pi4}2=\dfrac{\sqrt2+1}{2\sqrt2}=\dfrac{2+\sqrt2}4$

$\implies \cos\dfrac{9\pi}8=\pm\sqrt{\dfrac{2+\sqrt2}4}$

Because $\pi<\dfrac{9\pi}8<\dfrac{3\pi}2$ (third quadrant), and $\cos x<0$ for $x$ in this interval, you must take the negative root, so

$\cos\dfrac{9\pi}8=-\sqrt{\dfrac{2+\sqrt2}4}=-\dfrac{\sqrt{2+\sqrt2}}2$